Kirchhoff's circuit laws
Kirchhoff's laws are two that deal with the and (commonly known as voltage) in the of s. Widely used in , they can be applied in time and frequency domains and form the basis for . Kirchhoff's current law (KCL) This law is also called '''Kirchhoff's first law', Kirchhoff's point rule, or Kirchhoff's junction rule (or nodal rule). This law states that, for any node (junction) in an , the sum of s flowing into that node is equal to the sum of currents flowing out of that node; or equivalently: The algebraic sum of currents in a network of conductors meeting at a point is zero. Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be succinctly stated as: : \sum_{k=1}^n {I}_k = 0 where n'' is the total number of branches with currents flowing towards or away from the node. The law is based on the conservation of charge where the charge (measured in coulombs) is the product of the current (in amperes) and the time (in seconds). If the net charge in a region is constant, the KCL will hold on the boundaries of the region. This means that KCL relies on the fact that the net charge in the wires and components is constant. Uses A version of Kirchhoff's current law is the basis of most , such as . Kirchhoff's current law is used with to perform . KCL is applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear. Kirchhoff's voltage law (KVL) This law is also called '''Kirchhoff's second law', Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule. This law states that The directed sum of the (voltages) around any closed loop is zero. Similarly to KCL, it can be stated as: : \sum_{k=1}^n V_k = 0 Here, n is the total number of voltages measured. Generalization In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of (which is one of ). This has practical application in situations involving " ". Example Assume an electric network consisting of two voltage sources and three resistors. According to the first law we have : i_1 - i_2 - i_3 = 0 \, The second law applied to the closed circuit ''s''1 gives : -R_2 i_2 + \mathcal{E}_1 - R_1 i_1 = 0 The second law applied to the closed circuit ''s''2 gives : -R_3 i_3 - \mathcal{E}_2 - \mathcal{E}_1 + R_2 i_2 = 0 Thus we get a in i_1, i_2, i_3 : : \begin{cases} i_1 - i_2 - i_3 & = 0 \\ -R_2 i_2 + \mathcal{E}_1 - R_1 i_1 & = 0 \\ -R_3 i_3 - \mathcal{E}_2 - \mathcal{E}_1 + R_2 i_2 & = 0 \end{cases} Which is equivalent to : \begin{cases} i_1 + (- i_2) + (- i_3) & = 0 \\ R_1 i_1 + R_2 i_2 + 0 i_3 & = \mathcal{E}_1 \\ 0 i_1 + R_2 i_2 - R_3 i_3 & = \mathcal{E}_1 + \mathcal{E}_2 \end{cases} Assuming : R_1 = 100\Omega,\ R_2 = 200\Omega,\ R_3 = 300\Omega : \mathcal{E}_1 = 3\text{V}, \mathcal{E}_2 = 4\text{V} the solution is : \begin{cases} i_1 = \frac{1}{1100}\text{A} \\6pt i_2 = \frac{4}{275}\text{A} \\6pt i_3 = - \frac{3}{220}\text{A} \end{cases} i_3 has a negative sign, which means that the direction of i_3 is opposite to the assumed direction i.e. i_3 is directed upwards. References Category:Electronics